Let {X,} be an independent Bernoulli sequence with parameter p, and let np, A < o as n oo. Verify directly that the Lindeberg condition fails for {X„}.

Q: Find a divergent sequence {an} such that {a2n} converges

A: Let us take: {an}= {-1, 1, -1, 1, -1, 1, -1, .......} This is an alternating series. So it diverges.…

Q: 30. а, 2 1+ %3D - n

A:

Q: Exercise 26. Suppose {an} is a sequence which converges to 0, {n} is another sequence, and s, k, and…

A:

Q: Find the explicit formula of the following sequence. 8 (72, 24, 8, 3, ...} O a O b Oc - Od ƒ(n) = a₁…

A: Given sequence is 72, 24, 8, 83, ...

Q: 7. Let {an1 and {bn}1 be convergent sequences with limit A and 0, respectively. If bn 0 for all n N…

A:

Q: 16. Show that if (s} and (ta} Cauchy sequence. Cauchy sequences, then {s + tn} is also are a.

A: Consider the provided question, We have to show that if {sn} and {tn} are Cauchy sequence then {sn +…

Q: Suppose that the sequence x0, x₁, x2... is defined by x = 6, x₁ = 3, and xk+2 = 2xk+1+3xk for k≥0.…

A:

Q: Let F denote the kth number in the Fibonacci sequence. (a) Prove that F2+1 − F²? = Fk−1Fk-2 - (b)…

A:

Q: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

Q: Prove that if a sequence (an)1 is not bounded, then there is a subsequence (akn)-1 which either…

A:

Q: Let {an} and {bn} be two Cauchy sequences of real numbers. Then show that the sequence {|an − bn|}…

A:

Q: Given a₁ = an ◆ Σ(14-k)= Select one: O a. 14,309 O b. 14,331 O C. 14,329 O d. 14,336 Σ(14--/k) Find…

A:

Q: Find the generating function of the sequence {a;}k20 defined by ak = -3 if k even; a = 3 if k odd.…

A:

Q: (a) Prove that, for each k E N, b - Ck = ak and b+ Ck = |ak]. (b) Prove that, if ak converges…

A:

Q: Describe an assumption that we are making about the distribution of the beans after “frolicking,”…

A: When marked beans are added back to the population in the cup, the researcher ask to 'frolic' or mix…

Q: Let {an} be a monotone sequence. Show that (a,} is a Cauchy sequence if and only if it is bounded.

A:

Q: 13. Let (an)=I be a Cauchy sequence, and let p : N → N be a one-to-one function. Show that the…

A: A sequence is called a Cauchy sequence if as the sequence progresses, the terms of the sequence…

Q: Let an be a sequence defined by a1 = 0 and a2 = 1 and for n > 3 an = (n – 1)(an-1+an-2). %3D

A: Here, the terms given in the series are, a1=0,a2=1an=n-1an-1+an-2 Calculate the third, forth and…

Q: (b) Let (an) and (b) be Cauchy sequences. Prove from first principles that their (an + bn) is a…

A:

Q: Use the definition to prove that {(-

A:

Q: Suppose that {a;} is a Cauchy sequence in a metric space X, d and lim an = p. Suppose, in addition,…

A: We have to prove that limn→∞bn=p Given that ai is a Cauchy sequence. Let ε&gt;0 then ∃N1∈N such that…

Q: Show directly from the definition that the sequence {xn}, defined by în Cauchy sequence. = 3n+2 n+1'…

A: Definition of Cauchy sequence xn of real numbers: ∀ε&gt;0 ∃ n∈N  such that |xn-xm|&lt;ε ∀m&gt;n

Q: 1. Let a, = 1 and an+1 = 1+ 1+an (a) Find the first six terms of the sequence {an}. (b) Prove that…

A: As you asked multiple sub parts so I answered 1st three subpart.

Q: uppose that the sequence x0, x1, x2... is defined by xo = 3, x₁ =0, x₂=5, and xk+3 = 13xk+1+12x for…

A:

Q: 33. Use the Squeeze Theorem to prove (sinn/n} c) {3"/n!} b) fn/2"} f by contradiction. a)

A: 33.) use squeeze theorem to prove that each of the following sequences converges a) sinnn  , (b)…

Q: Five people (Alex, Byron, Chloe, Dexter, and Emily) form a club, N = {A, B, C, D, E). Chloe and…

A: Five people (Alex, Byron, Chloe, Dexter and Emily) form a club N = { A,B,C,D,E }

Q: a1 = 1, an+1 = 3 - An induction method to prove monotor

A:

Q: Suppose f: D→ R with ao the only accumulation point of D. If {an}1 is a Cauchy sequence with an ED…

A:

Q: an 57" n! nal

A: